UNIFORMLY QUASIREGULAR MAPPINGS
ON ELLIPTIC RIEMANNIAN MANIFOLDS
Riikka Kangaslampi
Department of Mathematics and Systems Analysis
Helsinki University of Technology
[email protected]
Matematiikan päivät 2010
Jyväskylä, 3.1.2010
Main result
Theorem
If there exists a uniformly quasiregular mapping on a compact
Riemannian manifold M n , there exists a quasiregular mapping
g : Rn → M n . In other words, the manifold M n is quasiregularly
elliptic.
A three-dimensional compact Riemannian manifold M 3 is elliptic if
and only if there exists a uniformly quasiregular mapping on M 3 .
Outline
1 Definitions
2 Rescaling principle and its consequences
3 3-dimensional elliptic manifolds
4 Follow-up
Outline
1 Definitions
2 Rescaling principle and its consequences
3 3-dimensional elliptic manifolds
4 Follow-up
Quasiregular mappings
n
n
Let D ⊂ R be a domain and f : D → R a non-constant mapping
1,n
of the Sobolev class Wloc
(D). We consider only
orientation-preserving mappings, which means that the Jacobian
determinant Jf (x) ≥ 0 for a.e. x ∈ D. Such a mapping is said to
be K-quasiregular, where 1 ≤ K < ∞, if
max |f 0 (x)h| ≤ K min |f 0 (x)h|
|h|=1
|h|=1
for a.e. x ∈ D, when f 0 is the formal matrix of weak derivatives.
Quasiregular mappings on manifolds
We generalize the definition to Riemannian manifolds with the help
of bilipschitz-continuous coordinate charts:
Let M and N be n-dimensional Riemannian manifolds. A
non-constant continuous mapping f : M → N is K -quasiregular if
for every ε > 0 and every m ∈ M there exists bilipschitz-continuous
charts (U, ϕ), m ∈ U, and (V , ψ), f (m) ∈ V , so that the mapping
ψ ◦ f ◦ ϕ−1 is (K + ε)-quasiregular.
A non-constant quasiregular mapping can be redefined in a set of
measure zero such that the mapping is made continuous, open and
discrete.
Uniformly quasiregular mappings
Let M be a compact Riemannian manifold. A non-injective
mapping f from a domain D ⊂ M onto itself is called uniformly
quasiregular (uqr) if there exists a constant 1 ≤ K ≤ ∞ such that
all the iterates f k are K -quasiregular.
We will assume our uniformly quasiregular mappings to be
non-injective. In other words, we assume the branch set
Bf = {x ∈ M | f is not locally homeomorphic at x}
to be non-empty.
Fatou and Julia sets
We define the Fatou set of a uqr mapping f : M → M as
Ff = {x ∈ M :
there exists an open set U ⊂ M such that x ∈ U
and the family {f k | k ∈ Z+ }|U is normal}.
The Julia set of the mapping f is Jf = M \ Ff .
By the definition, Fatou sets are open, and therefore Julia sets are
closed. Both Fatou and Julia set are completely invariant.
Theorem
Let f : M → M be a uqr mapping, with deg(f ) ≥ 2, on a compact
Riemannian manifold M. Then the Julia set Jf of the mapping f
is non-empty.
Outline
1 Definitions
2 Rescaling principle and its consequences
3 3-dimensional elliptic manifolds
4 Follow-up
Rescaling principle on a manifold
Theorem
Let F be a family of K -quasiregular mappings f : Ω → M n , where
M n is a closed compact Riemannian manifold, and Ω ⊂ Rn a
domain. If the family F is not equicontinuous at a point a ∈ Ω,
there exists a sequence of real numbers rj & 0, a sequence of
points aj → a, a sequence of mappings {fj } ⊂ F and a
non-constant K -quasiregular mapping h : Rn → M n such that
fj (rj x + aj ) → h(x)
locally uniformly in Rn . Especially M n is K -quasiregularly elliptic.
Manifolds supporting uqr mappings are elliptic I
We use the rescaling principle to prove the first part of our main
theorem.
Let f : M n → M n be a K 0 -quasiregular mapping on a smooth,
oriented, compact Riemannian n-manifold M, with a non-empty
branch set. Let x0 ∈ Jf , and let ϕ : U → Rn be such a
L-bilipschitz-continuous coordinate mapping in some
neighbourhood U of the point x0 that ϕ(x0 ) = 0 and
ϕ(U) = B(0, 1).
Define a composite mapping
fν := f ν ◦ ϕ−1 |B(0,1) : B(0, 1) → M n
from the iterates f ν of f and the coordinate mapping ϕ.
Manifolds supporting uqr mappings are elliptic II
All the mappings fν , ν ∈ N, are K -quasiregular with the same
constant K = K (K 0 , L).
The family of mappings F = {fν | ν ∈ N} is not normal, since
x0 = ϕ−1 (0) ∈ Jf , meaning that 0 ∈ JF .
Starting with a uqr mapping f : M → M we have thus constructed
a family F = {fν | ν ∈ N} of K -quasiregular mappings
fν = f ν ◦ ϕ−1 |B(0,1) : B(0, 1) → M,
and the family F is not normal at the origin. Now we can use the
rescaling principle, and as a limit mapping we get a K -quasiregular
mapping g : Rn → M.
Outline
1 Definitions
2 Rescaling principle and its consequences
3 3-dimensional elliptic manifolds
4 Follow-up
3-dimensional elliptic manifolds
Thurston’s conjecture states that after a three-manifold is splitten
into its connected sum and the Jaco-Shalen-Johannson torus
decomposition, the remaining components each admit exactly one
of the following model geometries: S3 , R3 , H3 , S2 × R, H2 × R,
f 2 (R), Nil, or Sol. (G. Perelman 2002)
SL
By J. Jormakka (1988), the only closed elliptic 3-manifolds are
those manifolds which are covered by S3 , S2 × R or R3 .
We already know that manifolds covered by S3 support uqr
mappings (K. Peltonen 1999). The Euclidean space forms and
manifolds covered by S2 × R are considered in the thesis.
Lattès-type uqr mappings I
Let Υ be a discrete group of isometries of Rn . A mapping
h : Rn → M is automorphic with respect to Υ in the strong sense
if
1
h ◦ γ = h for any γ ∈ Υ,
2
Υ acts transitively on the fibres Oy = h−1 (y ).
By the latter condition we mean that for any two points x1 , x2 with
h(x1 ) = h(x2 ) there is an isometry γ ∈ Υ such that x2 = γ(x1 ).
Lattès-type uqr mappings II
Theorem
Let Υ be a discrete group such that h : Rn → M is automorphic
with respect to Υ in the strong sense. If there is a similarity
A = λO, λ ∈ R, λ 6= 0, and O an orthogonal transformation, such
that
AΥA−1 ⊂ Υ,
then there is a unique solution f : h(Rn ) → h(Rn ) to the Schröder
functional equation
f ◦h =h◦A
and f is a uniformly quasiregular mapping.
Note that f k ◦ h = h ◦ Ak for all k.
3-dimensional Euclidean space forms I
By J. Wolf (1984) there are just 6 affine diffeomorphism classes of
compact connected flat 3-dimensional Riemannian manifolds. They
are represented by the manifolds R3 /Γ, where Γ is one of 6 groups.
We consider here the case G2 and the corresponding manifold M2
from the thesis as an example.
In the polyhedron schema for M2 we have a cube where we identify
opposite vertical faces and glue the top to the bottom of the cube
with a twist of angle π.
3-dimensional Euclidean space forms II
We obtain a Lattès-type uqr mapping on M2 as follows:
Construct a covering map g2 : T 3 → M2 such that T 3 covers
the manifold twice.
Define mapping F2 : R3 → R3 as F2 : x 7→ 3x.
F2 and covering map π1 induce a mapping F20 on the torus.
The mapping F2 descends to a Lattès-type uqr mapping
f2 : M2 → M2 .
F
R3 −−−2−→ R3



π
π2 y
y 2
F0
T 3 −−−2−→


g2 y
f
T3

g2
y
M2 −−−2−→ M2
Manifolds covered by S2 × R
There are only two orientable compact 3-manifolds which have
S2 × R as the Riemannian covering space: the sphere bundle
S2 × S and the connected sum of two projective 3-spaces P3 #P3 .
We take here P3 #P3 as an example.
The manifold P3 #P3 is obtained by identifying diametrical points
of the boundary spheres K1 and K2 of S2 × I .
Manifold P3 #P3 I
The dotted 2-sphere separates this manifold into two punctured
projective spaces. The fibres are the radii of S2 × I ; any two
diametrical radii form one fibre.
Manifold P3 #P3 II
Let us look at the projective space P3 #P3 as a block in R3 :
Manifold P3 #P3 III
We define a covering map g : R3 → P3 #P3 which takes each
one-by-one cube in R3 to P3 #P3 according to the previous picture.
We use the mapping F : R3 → R3 , where F : x 7→ 2x for any
x ∈ R3 . The mappings F and g again induce a mapping f to the
manifold:
F
R3 −−−−→ R3



g
gy
y
f
P3 #P3 −−−−→ P3 #P3
The mapping f is a well-defined and uniformly quasiregular
mapping of Lattès type.
Outline
1 Definitions
2 Rescaling principle and its consequences
3 3-dimensional elliptic manifolds
4 Follow-up
References and Follow-up
Kangaslampi, Riikka
Uniformly quasiregular mappings on elliptic Riemannian manifolds.
Ann. Acad. Sci. Fenn. Math. Diss. 151 (2008)
Since the thesis was published already in 2008, some progress has
been made after that:
Astola, L., Kangaslampi, R., and Peltonen, K.
Lattés-type mappings on compact manifolds
(preprint)
Kangaslampi, R., and Peltonen, K.
Uniformly quasiregular mappings on lens spaces
(in preparation)
Thanks!
Special thanks go to
Instructor Doc. Kirsi Peltonen
Supervisor Prof. Olavi Nevanlinna
Opponent Prof. Gaven Martin
Pre-examiners Prof. Aimo Hinkkanen and PhD Pekka Pankka